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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>





<a NAME="E1:61"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> ;<br/>
<a NAME="E2:61"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> 
 
;<br/>
<b>assume </b><a NAME="E3:61"/>
<font color="Maroon">A1</font> is <a href="prob_1.html#V2">non-increasing</a>
 ;<br/>

<a NAME="E4:61"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="prob_2.html#T14">PROB_2:14</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> </span>}</span> 
 ;<br/><a NAME="E2:61_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="setlim_1.html#T6" target="_self">Th6</a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E3:61_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:61"/>
<span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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